Chapter 24.1 - Proportionality in the size of A0, A1, A2 series papers

In the previous section we saw proportionality in the case of TV screens. In this section we will see Proportionality in the size of writing papers.

• We frequently use A4 size paper for writing, printing and taking photocopies. 
• This A4 is one of the several sizes which form a series. The other members of this series are A3, A2, A1 and A0. 
• A0, A1 and A2 are larger sizes. While A3 and A4 are smaller sizes. 
• All of them are rectangular in shape. 
• And all those rectangles are members of a single family: The '√2:1 rectangle family'
• From the name, the following information can be obtained:
    ♦ In all those rectangles, the ratio length : width is √2:1
• That means ^length⁄_width = ^√2⁄₁ ⇒ length = √2 × width
• So we can write: Which ever rectangle we take from that family, it's length will be √2 times the width.

We know that √2 is 1.414... (Details here). It is a non-recurring, non-terminating decimal. Why choose such a number for the ratio? We will see the answer soon:

1. Take any one member from the family. It is shown in the fig.24.5(a) below. Let it's length be a1 and width be b1. Since it is a member of the family, ^a1⁄_b1= √2

Fig.24.5

 2. Divide it into two equal parts by drawing a line through the centre and parallel to the width. This is shown in fig.b. 
3. Now fold the paper through this line. This is shown in fig.c. We will get a new rectangle. This new rectangle will have the following properties: 
• It's length will be equal to the width b1 of the original rectangle
• It's width will be equal to half of the length a1 of the original rectangle
4. Let the length of the new rectangle be a2 and it's width b2
We can write: a2 = b1 AND b2 = ^a1⁄₂
5. Now take the ratio ^length⁄_width of the new rectangle. We get:
^length⁄_width of the new rectangle = ^a2⁄_b2 = ^b1⁄_(a1/2) = ^2b1⁄_a1
6. But from (1) we get ^b1⁄_a1 = ¹⁄_√2. Substituting this in (5) we get:
^length⁄_width of the new rectangle = ²⁄_√2  = ^(√2×√2)⁄_√2 = √2

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So we can write:

• We take any one rectangle from the '√2:1 rectangle family'
• It will have ^length⁄_width ratio = √2:1
• We make that rectangle into exact half by folding it width wise
• The new rectangle so obtained will also have ^length⁄_width ratio = √2

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The above property was first discovered in Germany. The A0, A1, A2... series is based on this property. It is shown in fig.24.6 below. 

Fig.24.6

• Fig.24.6(a) shows an A0 size paper. Fig.b shows two A1 size papers. We can see that two A1 sheets make one A0 sheet. In the same way, two A2 sheets make one A1 sheet ans so on...
• Also width of A1 is equal to half the length of A0. Length of A1 is the width of A0 and so on...
■ This series is now accepted as the standard in most parts of the world. So let us take a closer look at the series:
1. The area of the A0 sheet, which is top most in the series, is fixed as 1 m². So what is it's length and width? We can find them easily:

(i) Let the length be l and width be w
(ii) Then Area = 1 m² = l × w
(iii) We know that l = √2×w
(iv) So we get 1 m² = l×w = √2×w×w = √2×w²

⇒ w² = ¹⁄_√2 = 0.707106 ⇒ w = √(0.707106) = 0.8409 m = 84.09 cm = 84.1 cm = 841 mm

(v) So length = √2×w = 1.414 × 841 = 1189 mm
• So for A0 size, length = 1189 mm and width = 841 mm
2. Once we fix up A0, the others in the series are easy. We do not need calculations with √2 any more. All we need to do is halving. Let us see:
■ Length of A1 = width of A0 = 841 mm
    Width of A1 = Half the length of A0 = ¹¹⁸⁹⁄₂ = 594.5 mm
■ Length of A2 = width of A1 = 594 mm
    Width of A2 = Half the length of A1 = ⁸⁴¹⁄₂ = 420 mm
■ Length of A3 = width of A2 = 420 mm
   Width of A3 = Half the length of A2 = ⁵⁹⁴⁄₂ = 297 mm
■ Length of A4 = width of A3 = 297 mm
   Width of A4 = Half the length of A3 = ⁴²⁰⁄₂ = 210 mm


■ The property that one sheet size can be made from halving the size just above, makes enlarging, reducing, preparation of brochures etc., easier.
■ Note: There will be infinite number of rectangles in the '√2:1 rectangle family'. But all of them do not become members of the A0, A1, A2... series. The 'chosen ones' are derived from A0 which has an area of 1 m².

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We will now see a solved example

Solved example 24.1
The area of A0 paper is 1m². Calculate the sides of an A4 paper correct to 1 mm, using a calculator
Solution:
Above, we have calculated the lengths of all papers in the series. But in this problem, the sides of only one size is asked. In such cases, there is an easier method.
1. Area of A0 size = 1
Area of A1 size = half of 1 = ¹⁄₂ m²
Area of A2 size = half of ¹⁄₂ = ¹⁄₄ m²
Area of A3 size = half of ¹⁄₄ = ¹⁄₈ m²
Area of A4 size = half of ¹⁄₈ = ¹⁄₁₆ m²
2. Now we calculate the sides of A4 size:
(i) Let the length be l and width be w
(ii) Then Area = ¹⁄₁₆ m² = l × w
(iii) We know that l = √2×w
(iv) So we get ¹⁄₁₆ m² = l×w = √2×w×w = √2×w²
⇒ w² = ¹⁄_(16√2) = 0.04419 ⇒ w = √(0.04419) = 0.2102 m = 210.2 mm 
(v) So length = √2×w = 1.414 × 210.2 = 297.3 mm

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In the next section we will see proportionality in some more cases.

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